Ultra Wideband Transient Scattering and Its Applications to Automated Target Recognition

Abstract

Reliable radar target recognition has long been the holy grail of electromagnetic sensors. Target recognition based on the singularity expansion method (SEM) uses a time-domain electromagnetic signature and has been well studied over the last few decades. The SEM describes the late time period of the transient target signature as a sum of damped exponentials with natural resonant frequencies (NRFs). The aspect-independent and purely target geometry and material-dependent nature of the NRF set make it an excellent feature set for target characterization. In this chapter, we aim to review the background and the state of the art of resonance-based target recognition. The theoretical framework of SEM is introduced, followed by signal processing techniques that retrieve the target-dependent NRFs embedded in the transient electromagnetic target signatures. The extinction pulse, a well-known target recognition technique, is discussed. This chapter covers recent developments in using a polarimetric signature for target recognition, as well as using NRFs for subsurface sensing applications. The chapter concludes with some highlights of the ongoing challenges in the field.

1. Introduction

The need to quickly and accurately identify enemies in confrontational situations is essential to most defense applications. Such decisions often rely upon radar target recognition. The two primary functions of radar are inherent in the acronym, whose letters stand for radio detection and ranging. There are two main categories of radar target recognition techniques: imaging and what is termed signature recognition. Imaging radars provide a visualization of the target using techniques such as focus spot scanning and inverse synthetic aperture [1]. Signature recognition radar extracts some characteristics or a feature set that characterizes the target. Some of the techniques, such as radar cross section (RCS) [2], polarization techniques [3], high-resolution range profiles (HRRP) [4], scattering centers [5], and multiple frequency measurements [6], are all under this category. The main drawback of these techniques is that the extracted parameters usually vary with incident aspect. For most radar target recognition problems, usually, the incident aspect angles of the target are not known a priori. It is, therefore, preferable to implement a technique that is purely dependent on the target itself and independent of its aspect to the radar.

One of the methods that overcome the aspet-dependent limitation is termed resonance-based target recognition [7]. As the name states, resonance-based target recognition essentially characterizes the radar target based on the natural resonant frequencies (NRFs) embedded in the target response. These NRFs are purely dependent on the physical attributes of the radar target, i.e., its dielectric properties and physical geometry, and these parameters are independent on the incident aspect [7] and incident polarization states [8]. Provided of course the resonances are well excited, the feasibility of using the NRFs for target recognition has been successfully demonstrated in the literature [9, 10, 11].

This chapter aims to provide an overview of the fundamentals and the development of resonance-based target recognition. We commence the chapter with a short discussion on RCS—a well-known frequency-domain method on how electromagnetic scattering is characterized at microwave frequencies. The singularity expansion method (SEM), which is the theoretical framework for resonance-based target recognition, provides the physical description of the transient scattering phenomena in time and frequency domains. Technical solutions for retrieving the target-dependent NRFs from the transient target signatures and automated target recognition (ATR) algorithms will be covered. The recent development including the prospect of using full polarimetric signatures, as well as the potential use of the techniques in other sensing applications, is also discussed. Comments and suggestions for future development are considered in conclusion.

2. Resonance-based target recognition

RCS is a well-known technique to characterize target from 3 MHz up to 300 MHz (HF to mm) [2]. Attempts have also been made to perform RCS measurement in the terahertz frequency region [12, 13]. RCS is a measure of the power that is returned or scattered in a given direction, normalized with respect to the power of the incident field. Mathematically, the RCS ($\sigma$) of a target is defined as [2]

$$\sigma =4\pi \underset{R\to \infty }{\text{lim}}{R}^2\frac{\left|{\mathbf{E}}_{\mathbf{s}}\right|}{\left|{\mathbf{E}}_{\mathbf{i}}\right|},$$E1

where $R$ is the range from the radar to the target and $E_i$ and $E_s$ are the amplitudes of the incident electric field from the radar transmitter and the scattered electric field from the target, respectively. The scattered field and thus the RCS of a definitive target, in general, vary as a function of incident aspects, receiving aspects, and excitation frequency. In the resonant regime which the wavelength of illumination ($\lambda$) and the target size ($L$) are comparable ($0.4\lambda \le L\le 10\lambda$), every part of the target affects every other parts. The total field of any part of the target is the vectorial sum of the incident and scattered field due to every part of the scattering body. This collective interaction determines the overall electromagnetic current induced on the target, which explains why the induced current at resonant frequencies is dependent on the physical attributes of the target.

Resonance-based target recognition is based on the resonating electromagnetic current induced on the target such that the ratio of excitation wavelength ($\lambda$) and the target size ($L$) has to be within the resonance scattering regime. The induced current serves as a secondary source and reradiates such that the resonant modes are embedded in the scattered target response. Rather than a frequency-domain characterization using RCS, one can also illuminate the target at a particular aspect in the time domain through a short-pulse excitation, measure the corresponding transient response at the same position (monostatic) or any another aspect (bistatic), and obtain the impulse response of the target. If one applies a Fourier transform to the target impulse response, this is the same as obtaining the frequency-domain “transfer function” of the target, thus evaluating the frequency dependency of the RCS at a particular transmit-receive configuration. Indeed, we assume that the bandwidth of the pulse is wide enough such that it covers at least the first few dominant resonant frequencies of the target.

2.1. Singularity expansion method

In the context of linear time-invariant (LTI) systems, the impulse response characterizes the behavior of the system and circuit. In the mid-1960s, Kennaugh and Moffatt [14] extended the concept of the impulse response and applied it to transient electromagnetic scattering from definitive targets. In the early 1970s, Baum [7] introduced the SEM that describes transient scattering phenomena. According to SEM, the entire transient target signature can be divided into two parts, namely, the early time and the late time period. Conceptually, the early time period is defined as the period from when the electromagnetic pulse initially strikes the target until the target is fully illuminated, while the commencement of the late time period is the time when global resonance has been fully established and the excitation is no longer present.

In the early time, the target is partly illuminated such that the majority of the scattering events originate locally from different parts of the target: specular reflections, diffraction from non-planar surfaces, edges, and corners. The occurrence of these events depends on when the pulse strikes on these edges and corners, and thus these early events are local, aspect dependent, and polarization dependent. They can be individually treated as scattering centers [5] or more generally using a wavefront description [15]. In the time domain, these scattering events correspond to impulse-like components, and they are time-varying (nonstationary) and can be modeled in a circuit context using an entire function [7, 16, 17].

In contrast, the commencement of the late time period is the time when the target is fully illuminated such that resonant modes at distinct frequencies are fully established. As previously mentioned the total field at any part of the target is a collective interaction of the incident and scattered field from every single part of the target. At particular frequencies, the induced current on the target is freely resonating, and resonant modes are established. These resonant modes, known as natural resonance frequencies (NRFs), are theoretically dependent only on the physical attributes of the target and are independent of the aspect [7] and polarization [8]. This allows them to be an excellent candidate to be used as a feature set for target classification. Mathematically, the late time target signature can be modeled as a sum of damped exponentials with constant residues, i.e.,

$$r\left(t\right)=\sum _{n=1}^N\left[A_n{e}^{s_{n}t}+A_n^{\ast }{e}^{s_n^{\ast }t}\right],t>T_l$$E2

where N is the modal order of the signal—the number of modes embedded in the late time response. It is assumed that only N modes are excited given that the pulse excitation is band-limited. $T_l$ is the onset of the late time and * denotes the complex conjugate. $A_n$ is the aspect-dependent residues. $s_n={\sigma }_n\pm j{\omega }_n$ is the complex NRF. ${\sigma }_n\left(<0\right)$ and ${\omega }_n$ are the damping coefficients and resonant frequencies of the n^th mode, respectively.

2.2. Resonance extractions for target classification

Target classification and recognition rely on accurate extraction of the target-dependent NRFs. There are two main approaches to extract these NRFs. The first one is based on the mathematical formulation of the scattering problem. Through a Fourier transform, the short-pulse excitation corresponds to a broadband of frequencies in the frequency domain. If we formulate the entire scattering problem at a particular frequency via an integral equation formulation, the integral equation relationship can be written in a matrix form and solved via moment methods [18]. In general, this can be written as

$$\left[\mathbf{Z}\right]\left[\mathbf{I}\right]=\left[\mathbf{V}\right],$$E3

where $\left[\mathbf{Z}\right]$ is an impedance-like matrix corresponding to the target geometry and $\left[\mathbf{I}\right]$ and $\left[\mathbf{V}\right]$ are column vectors corresponding to the unknown current induced on the target and the excitation, respectively. The natural resonant modes of the target, $s_n={\sigma }_n\pm j{\omega }_n$, are those such that the homogenous version of Eq. (3) has a nontrivial solution. This implies that the solution exists when the determinant of the $\left[\mathbf{Z}\right]$ matrix equals zero, thus establishing the singularities of the $\left[\mathbf{Z}\right]$ matrix. These singularities can be extracted using a typical (complex) root searching method such as Muller’s method [7]. The $\left[\mathbf{Z}\right]$ matrix is constructed based solely on the geometry and dielectric properties of the target. The NRFs are therefore totally independent of the incident aspect angle. A major limitation is that the entire physical problem needs to be modeled using moment method, and each $\left[\mathbf{Z}\right]$ corresponds to only one frequency. This means that one needs to repeat the entire root searching process for all the frequency points, which requires extensive computation. Another limitation is that this method cannot be applied directly to measured data.

Shortly after the proposition of SEM by Baum, Van Blaricum and Mittra [19] proposed using Prony’s method which directly retrieves the NRFs and residues from the late time response in a computationally affordable manner. Prony’s method addresses the above limitations, and it has drawn significant attention in the field. However, the primary drawback of Prony’s method is that the accuracies of the extracted parameters are highly sensitive to noise and the estimated modal order of the signal. Since then, many techniques have been proposed which can accurately retrieve the modal order, NRFs, and residues with noisy target signatures. To date, direct extraction of NRFs from late time target signature has become the principal approach in this context, and matrix pencil methods (MPM) developed by Sarkar and Pereira [20] have become the main tool for this purpose.

To illustrate how NRFs can be used for target classification, as well as the aspect dependency nature of the resides, an example of a 1 m wire target illuminated under different excitation aspect angles $\theta$ shown in Figure 1 is considered [21]. The wire target of length ($\mathrm{\ell }$) and radius ($a$) ratio $\mathrm{\ell }/a=200$ is excited by a plane wave with the electric field in the plane of the wire. The transient target responses are obtained using the indirect time-domain method [22]—the scattering problem is first solved in the frequency domain at a large number of discrete frequency points, followed by Gaussian windowing to model a Gaussian pulse excitation [23] and an inverse Fourier transform. Here, the electromagnetic problem is solved using the moment method solver FEKO [24] from 4.39 MHz to 9 GHz with 2048 equally spaced samples. Given the target and incident aspect angles to the target, the late time can be approximated by

$$T_l=T_b+T_p+2T_{\mathit{tr}},$$E4

where $T_{\mathit{tr}}$ is the maximal transit time of the target, $T_p$ is the effective pulse duration, and $T_b$ is the estimated edge when the pulse strikes the leading edge of the target [11]. The Gaussian pulse commences at $T_b=10\mathit{ns}$ with $T_p=0.22\mathit{ns}$. According to the geometry of Figure 1, $T_{\mathit{tr}}=\mathrm{\ell }\text{cos}\theta /c$ where $c=3\times {10}^{8}m/s$, and the excitation angles of $\theta ={15}^{°},{45}^{°}\text{and}{75}^{°}$ are considered, resulting in $T_l\approx 16.6\mathrm{ns},14.92\mathrm{ns}\text{and}11.9\mathrm{ns}$, respectively. The NRFs are extracted using the MPM [20] with late time samples from 17 to 140 ns. The first ten dominant extracted NRFs are listed in Table 1 and are compared with the ground truth—NRFs extracted using a root searching procedure of the $\left[\mathbf{Z}\right]$ matrix [25].

Figure 1.

The wire scatterer with plane wave incidence (reprinted from [21] with permission from IEEE).

$n$	f (MHz)	$\left[\mathbf{Z}\right]$ [25]	Matrix pencil method [20]
		$s_{n}L/c$	${15}^{°}$, $s_{n}L/c$	${45}^{°}$, $s_{n}L/c$	${75}^{°}$, $s_{n}L/c$
1	138	$-0.260\pm j2.91$	$-0.252\pm j2.87$	$-0.253\pm j2.87$	$-0.252\pm j2.87$
2	286	$-0.381\pm j6.01$	$-0.372\pm j5.93$	$-0.370\pm j5.93$	$-0.373\pm j5.93$
3	432	$-0.468\pm j9.06$	$-0.455\pm j9.01$	$-0.458\pm j9.01$	$-0.444\pm j9.05$
4	583	$-0.538\pm j12.2$	$-0.525\pm j12.1$	$-0.512\pm j12.1$	$-0.545\pm j12.1$
5	730	$-0.600\pm j15.3$	$-0.585\pm j15.2$	$-0.609\pm j15.2$
6	879	$-0.654\pm j18.4$	$-0.637\pm j18.3$	$-0.833\pm j18.4$	$-0.881\pm j17.6$
7	1027	$-0.704\pm j21.5$	$-0.692\pm j21.4$		$-0.850\pm j21.6$
8	1175	$-0.749\pm j24.6$	$-0.733\pm j24.6$	$-1.043\pm j24.5$^
9	1323	$-0.792\pm j27.7$	$-0.785\pm j27.7$	$-0.732\pm j27.9$	$-1.005\pm j28.6$
10	1471	$-0.832\pm j30.8$	$-0.817\pm j30.8$	$-1.294\pm j30.9$^

Table 1.

Comparison between the extracted NRFs ($s_{n}L/c$) using $\left[\mathbf{Z}\right]$ matrix and MPM (reprinted from [21] with permission from IEEE).

The results show that all the first ten dominant resonant modes can be extracted only at $\theta ={15}^{°}$, while some modes cannot be retrieved at $\theta ={45}^{°}$ and $\theta ={75}^{°}$. To investigate of what is happening as the incidence angle varies, we transform the transient signatures to the joint time-frequency (TF) domain such that the existence and occurrences of the NRFs and scattering phenomena are clearly observed. Of all the time-frequency distributions (TFDs) in the Cohen class and the reassigned TFDs [26, 27], the Smooth Pseudo Wigner-Ville Distribution (SPWVD) gives reasonable TF resolutions without introducing uninterpretable artifacts [28, 29, 30].

The SPWVD, together with the corresponding time and frequency responses of the three signals, is shown in Figure 2(a)–(c). At $\theta ={15}^{°}$, all the ten modes are identified in the joint TF domain. In Figure 2(b), modes 1 to 5 and mode 9 are observed when the incident angle is changed to ${45}^{°}$. Mode 6 is not observed due to its small residue. Two resonant modes at about 1.2 and 1.5 GHz are also found in the figure. They have similar frequencies to modes 8 and 10 but with higher damping factors (marked with ^ in Table 1), which could probably be the higher layer NRFs [31]. As the incident angle changes to ${75}^{°}$, only modes 1 to 4, 6, 7, and 9 are excited, and they are be found correspondingly to those frequencies in Figure 2(c). In addition to the TF results, the NRFs also appear as peaks in the frequency response. As shown in both TF and frequency domains, it is apparent that the strength of the resonant modes, i.e., the residues, changes as the incident aspect varies, which validates the aspect dependency nature of the residues.

Figure 2.

SPWVD of the wire scatterer with plane wave incidence at (a) $\theta ={15}^{°}$, (b) $\theta ={45}^{°}$, and (c)$\theta ={75}^{°}$ (reprinted from [21] with permission from IEEE).

2.3. Extinction pulse technique

The target-dependent nature of the NRFs implies that the NRF patterns appearing in the S-plane are unique for a given target. Target recognition can thus be easily achieved by visually inspecting the patterns of the NRFs in the S-plane [32]. To automate the identification procedure, Rothwell [9, 10, 11] proposed that target recognition can be performed by convolving the target signatures with a special type of filter, known as the “extinction pulse” or the “E-pulse,” in the time domain. The E-pulse is specially designed such that it will annul all the NRFs embedded in the late time response only if it is convolved with the response from the “true target.” Mathematically, the E-pulse, $e\left(t\right)$, can be defined as [9, 10, 11]

$$c\left(t\right)=\underset{0}{\overset{T_E}{\int }}e\left(\tau \right)r\left(t-\tau \right)\mathit{d\tau }=0\text{for}t>T_L=T_l+T_E,$$E5

where $r\left(t\right)$ is the late time target signature defined in Eq. (2), $T_l$ is the commencement of the late time period, and $T_E$ is the duration of $e\left(t\right)$.

To illustrate how E-pulse operates, an ATR scenario is shown in Figure 3 [11]. The goal here is to identify the target given a measured target signature $v\left(t\right)$ from an unknown target [11]. In the target library, a number of E-pulses corresponding to different targets are stored. To perform ATR, $v\left(t\right)$ is convolved with each of the E-pulses independently. According to Eq. (5), the one with a null response or smallest signal strength corresponds to the “true” target. Target identification is thus performed by monitoring the output of the convolution and picks up the one with the small energy level. To automate the process, the output is quantified using E-pulse discrimination number, $\mathit{EDN}$, and the E-pulse discrimination ratio, $\mathit{EDR}$, which are defined as follows [11]:

$$\mathit{EDN}=\left[\underset{T_L}{\overset{T_L+W}{\int }}{\left(e{\left(t\right)}^{\ast }v\left(t\right)\right)}^2\mathit{dt}\right]{\left[\underset{0}{\overset{T_e}{\int }}{e}^2\left(t\right)\mathit{dt}\right]}^{-1}$$E6

$$\mathit{EDR}=10{log}_{10}\left[\frac{\mathit{EDN}}{\text{min}\left\{\mathit{EDN}\right\}}\right]$$E7

Figure 3.

Automated target recognition using the E-pulse technique [11]. The goal is to determine which target does the unknown target signature $v\left(t\right)$ correspond to. $v\left(t\right)$ is convolved with all the E-pulses in the target library, and the corresponding EDN and EDR values are computed. the E-pulse that results in minimum value of EDN or equivalently 0 dB of EDR indicates the E-pulse “matches” with $v\left(t\right)$—The true target is thus the one that generate this E-pulse.

Therefore, the target signature yielding the smallest $\mathit{EDN}$ and 0 dB $\mathit{EDR}$ is the one corresponding to the target of interest. This forms the basis for E-pulse ATR.

In most of our studies, we know a priori which one is the “right” target, and our goal is to determine if E-pulse is capable of discriminating the targets in new applications; for instance, the “banded” E-pulse technique that better discriminates between similar targets [33], a novel technique for subsurface target detection [34], and ATR using polarimetric signatures (Section 3.1). Figure 4 shows the flowchart of how we validate the E-pulse technique. For the case of three targets, there are three target signatures and three E-pulses, resulting in nine convolutions. Instead of using $\mathit{EDN}$ and $\mathit{EDR}$ in Eqs. (6) and (7), we modify them and introduce ${\mathit{EDN}}_{p,q}$ and discrimination ratio, ${\mathit{DR}}_{p,q}$, to quantify and convolution outcome and discrimination performance. They are given as follows [33, 35]:

$${\mathit{EDN}}_{p,q}=\frac{\left[\underset{T_L}{\overset{T_L+W}{\int }}{\left(e_q{\left(t\right)}^{\ast }{r}_p\left(t\right)\right)}^2\mathit{dt}\right]}{\left[\underset{0}{\overset{T_e}{\int }}{e}_q^2\left(t\right)\mathit{dt}\right]\left[\underset{T_L}{\overset{T_L+W}{\int }}{r}_p^2\left(t\right)\mathit{dt}\right]}$$E8

$${\mathit{DR}}_{p,q}=10{\text{log}}_{10}\left[\frac{{\mathit{EDN}}_{p,q}}{{\mathit{EDN}}_{p,p}}\right].$$E9

Figure 4.

Validation of the E-pulse technique for new applications with known targets: (a) to (c) illustrate how the ${\mathit{EDN}}_{p,q}$ and ${\mathit{DR}}_{p,q}$ values are computed for the case of three known targets with target signatures ($r_1\left(t\right)$, $r_2\left(t\right)$, $r_3\left(t\right)$) and E-pulses ($e_1\left(t\right)$,$e_2\left(t\right)$,$e_3\left(t\right)$). As we know a priori of the “true” target, our goal is to test if we could obtain positive ${\mathit{DR}}_{p,q}$ ($p\ne q$) values.

${\mathit{EDN}}_{p,p}$ is the case when the E-pulse and target response are from the same target, while ${\mathit{EDN}}_{p,q}$ is the case which the E-pulse and target returns are from different targets. A positive value of ${\mathit{DR}}_{p,q}$ implies that ${\mathit{EDN}}_{p,q}$ is greater than ${\mathit{EDN}}_{p,p}$ and thus a successful target recognition. As we have a priori knowledge of correct targets, positive ${\mathit{DR}}_{p,q}$ values are simply a validation of the success of E-pulse ATR or detection in the aforementioned new applications.

In practice, the E-pulse of a target can be obtained by numerically solving the convolution integral in Eq. (5) given the target signature [36] or using the formulation in [10] given the NRFs of the target. The E-pulses used in our studies (e.g., [33, 34, 35]) are constructed using the formulation in [10] together with the NRFs extracted from MPM.

3. Recent developments

Upon the introduction of SEM in the 1970s until the mid-1990s, research has mainly concentrated on three directions: theoretical studies with better modeling and description of early time scattering phenomena [15, 16], signal processing solutions to better retrieve SEM parameters for target characterization [17, 19, 20, 36, 37, 38, 39], and development of E-pulse and other ATR solutions for target recognition [11, 33, 40, 41, 42, 43, 44]. Most studies focused on ATR for targets in free space. The E-pulse proposed by Rothwell et al. [9, 10, 11, 36, 42] and the MPM algorithm by Sarkar and Pereira [20] have become the benchmark for filter-based ATR techniques and NRF extraction in this context. Since the 2000s, research activities have shifted toward applying the techniques developed in resonance-based target recognition for different applications. These include subsurface target detection, nondestructive evaluation, and medical diagnosis. Instead of using the linearly polarized electromagnetic wave to excite the target, the prospect of using fully polarimetric target signatures for ATR has also been investigated.

In this section, we first discuss the use of full polarimetric measurement and its impact on resonance-based target recognition. The merit of using polarimetric transient signatures is demonstrated through numerical examples. Then, different strategies for handling the multiple-aspect multiple-polarization data sets in multi-static scenarios are evaluated. Toward the end of the section, research activities on applying the E-pulse technique for different applications will be covered.

3.1. ATR using polarimetric signatures

The aspect dependency, as demonstrated in the above wire scattering example, as well as the polarization dependency of the residues, would limit the reliability and performance of ATR as it is uncertain whether all the dominant NRFs are well excited because the target orientation is usually not known a priori. In the mid-2000s, Shuley et al. [9] studied the residues retrieved from a target at 18 linear polarization angles at the same aspect and found that the residues of some NRFs could be so small at some polarization angles such that these NRFs are not retrieved. This was the first instance where polarization dependency of the residues is demonstrated. If we evaluate the RCS of the target at the resonant frequencies, the amplitudes of the scattered field, in general, vary as the observation aspects and polarization. This explains the aspect and polarization dependencies of the residues.

To accurately characterize a target using NRFs extracted from measured target signatures, it is important to incorporate more than one target signature obtained from different aspects and polarization states. Lui and Shuley [45] investigated different ways to process target signatures obtained from a number of polarization angles at a single aspect. Our results show that using the modified MPM [39] that allows extraction of one set of NRFs from multiple target signatures is preferred as it does not corrupt the original time-domain information and the risk of ignoring dominant modes is eliminated [45]. The major drawback is that the computational load grows as the number of target signatures (aspects and polarization angles) is increased. Also, when linear polarization is used, it is not trivial to decide how many polarization angles are sufficient. To better understand the impact of using the different polarimetric response for ATR, examples of some simple wire targets will first be presented. Then, different ways to handle data set obtained from multiple-aspect multiple polarization measurements for target classification will be presented.

In polarimetry, the Sinclair scattering matrices [46] in the linear and circular polarization bases are given by

$$\left[S\left(H,V\right)\right]=\left[\begin{array}{ll}{S}_{\mathit{HH}}& S_{\mathit{HV}}\\ S_{\mathit{VH}}& S_{\mathit{VV}}\end{array}\right]\text{and}\left[S\left(L,R\right)\right]=\left[\begin{array}{ll}{S}_{\mathit{LL}}& S_{\mathit{LR}}\\ S_{\mathit{RL}}& S_{\mathit{RR}}\end{array}\right].$$E10

Here, the subscripts denote the polarization channels (sets of transmit-receive polarization), where V, H, L, and R correspond to vertical, horizontal, left-hand, and right-hand circular polarization, respectively. The scattering matrices relate the incident and scattered electric field of the target under different polarization bases at a particular frequency. The first and second subscripts in each of the term inside the Sinclair scattering matrix correspond to the polarization state from the transmitting and receiving antennas. For instance, $S_{\mathit{VH}}$ corresponds to the case where the target is illuminated using vertically polarized plane wave and the horizontally polarized scattered field is measured. Once the Sinclair scattering matrix is measured in an orthogonal basis, all the polarization states (linear, circular, and elliptical) can be synthesized such that it is not necessary to illuminate the target with a large number of linear polarization angles from the same aspect [9, 45]. Only four target signatures (three for monostatic) are required which considerably reduces the amount of data to be processed.

To investigate the impact of the excitation and receiving states to the performance of ATR, numerical examples of three wire targets shown in Figure 5 were studied [47]. The targets are made up of two wire segments—a vertical wire segment (main body) of 1 m and a horizontal wire segment of 0.3 m (Target 1 and Target 3) and 0.2 m (Target 2) located at the center (Target 1 and Target 2) and at 0.2 m away from the end of the main body (Target 3), respectively. The scattering problems of these targets are solved in the frequency domain using FEKO [24] with 512 equally spaced frequency samples from 3.9 MHz to 2 GHz. The polarimetric transient signatures, or equivalently the scattering matrices in the time domain, i.e.,

$$\left[S_{\left(H,V\right)}\left(t\right)\right]=\left[\begin{array}{ll}{S}_{\mathit{HH}}\left(t\right)& S_{\mathit{HV}}\left(t\right)\\ S_{\mathit{VH}}\left(t\right)& S_{\mathit{VV}}\left(t\right)\end{array}\right]\text{and}\left[S_{\left(L,R\right)}\left(t\right)\right]=\left[\begin{array}{ll}{S}_{\mathit{LL}}\left(t\right)& S_{\mathit{LR}}\left(t\right)\\ S_{\mathit{RL}}\left(t\right)& S_{\mathit{RR}}\left(t\right)\end{array}\right],$$E11

Figure 5.

Wire targets and corresponding excitation polarization references. (a) Target 1: $r=0.3m$, target 2: $r=0.2m$, and (b) target 3 (reprinted from [47] with permission from IEEE).

are determined using the aforementioned indirect time-domain method [22]. For each target, the NRFs from each target signature is extracted using MPM [20] resulting eight sets of NRFs and residues. At each polarization state, the E-pulses for each of the three targets (denoted as target q) are constructed using the extracted NRFs [10]. To evaluate the E-pulse ATR performance, the E-pulse validation procedures shown in Figure 4 are applied. The E-pulse of each target (denoted as target q) is convolved with the target signatures from different targets (denoted as target p) but with the same polarization state. Before the convolution, both the E-pulses and target signatures are resampled as usually the sampling rate of the E-pulse and the target signatures are not the same [35]. The ${\mathit{EDN}}_{p,q}$ and ${\mathit{DR}}_{p,q}$ are computed, resulting in nine sets of ${\mathit{EDN}}_{p,q}$s and ${\mathit{DR}}_{p,q}$s for each polarization state. The corresponding results are tabulated in Table 2.

${\mathit{DR}}_{p,q}$(dB)	(a) $S_{\mathit{VV}}\left(t\right)$			(b) $S_{\mathit{HH}}\left(t\right)$
	$p=1$	$p=2$	$p=3$	$p=1$	$p=2$	$p=3$
$q=1$	0	0.01	42.9	0	46.6	0.002
$q=2$	−0.01	0	42.9	126.2	0	126.2
$q=3$	65.4	65.4	0	−0.002	53.9	0
${\mathit{DR}}_{\mathit{p},\mathit{q}}$(dB)	(c) ${\mathit{S}}_{\mathit{LL}}\left(\mathit{t}\right)$			(d) ${\mathit{S}}_{\mathit{LR}}\left(\mathit{t}\right)$
	$\mathit{p}=\mathbf{1}$	$\mathit{p}=\mathbf{2}$	$\mathit{p}=\mathbf{3}$	$\mathit{p}=\mathbf{1}$	$\mathit{p}=\mathbf{2}$	$\mathit{p}=\mathbf{3}$
$q=1$	0	31.4	36.0	0	38.9	42.9
$q=2$	27.0	0	43.8	26.7	0	44.6
$q=3$	48.4	53.1	0	43.6	49.3	0

Table 2.

ATR using target signatures under different polarization bases using E-pulse technique with the corresponding ${\mathit{DR}}_{p,q}$ (dB) values (reprinted from [47] with permission from IEEE).

Under vertical excitation, the main body is excited but not the horizontal wire segment. Theoretically, the cross polarized response should be zero in this case and vice versa for horizontal polarization excitation, and thus we only consider the co-polarized components. As tabulated in Table 2, the E-pulse technique fails to recognize between Target 1 and Target 2 for the case of $S_{\mathit{VV}}\left(t\right)$, with ${\mathit{DR}}_{1,2}$ and ${\mathit{DR}}_{2,1}$ values near to 0 dB as only the NRFs corresponding to the main body are excited. For the case of $S_{\mathit{HH}}\left(t\right)$, the horizontal wire segments of the two targets are well excited, and ${\mathit{DR}}_{1,2}$ and ${\mathit{DR}}_{2,1}$ values of 46.6 and 126.2 dB are obtained, which indicates successful target recognition. However, almost 0 dB of ${\mathit{DR}}_{1,3}$ and ${\mathit{DR}}_{3,1}$ values result. This is because the length of the horizontal wire segment of Target 1 and Target 3 is identical and the transient responses are strongly dominated by the horizontal wire segments. Under vertical polarization, the current distributions of the two targets are different due to different positions of the horizontal wire segments. The ${\mathit{DR}}_{1,3}$ and ${\mathit{DR}}_{3,1}$ values of 42.9 and 65.4 dB result which indicate the E-pulses successfully distinguish between Target 1 and Target 3. With different positions and length of the horizontal wire segments, the E-pulse is capable of distinguishing between Target 2 and Target 3 for both $S_{\mathit{VV}}\left(t\right)$ and $S_{\mathit{HH}}\left(t\right)$. Under monostatic configuration and circularly polarized illumination, $S_{\mathit{LL}}\left(t\right)=S_{\mathit{RR}}\left(t\right)$ and $S_{\mathit{LR}}\left(t\right)=S_{\mathit{RL}}\left(t\right)$, and thus we only need to consider $S_{\mathit{LL}}\left(t\right)$ and $S_{\mathit{LR}}\left(t\right)$. ${\mathit{DR}}_{p,q}$ values ($p\ne q$) of at least 26 dB result in all cases. Such results indicate that using circularly polarized target signatures can successfully discriminate the three targets.

In this example, ATR performance under different polarization states is studied. When the target is illuminated under linear polarization, only the main body or the horizontal wire segment is excited (details of the extracted NRFs can be found in [47]). The E-pulses are constructed using the incomplete set of NRFs. The target is poorly characterized and is not fully illuminated—these are the two main causes of the inconsistency in ATR performance. An example of the inconsistency in ATR performance due to aspect dependencies of the NRFs is reported in [48]. When the target is illuminated under circular polarization, the NRFs of both wire segments are adequately excited. The constructed E-pulse contains the domain NRFs of the entire target. The target is well characterized and well illuminated under circular polarization. The consistent ATR performance originates from the fact that the NRFs of both wire segments of the targets are well excited. The findings from this example demonstrated the importance of including all the dominant NRFs (including both the global and partial/substructure resonances [30]) for target classification, as well as the importance of exciting all the dominant NRFs, especially when a library of similar targets is considered [33].

3.2. Target classification using multiple-aspect multiple-polarization data set

Owing to the aspect and polarization dependencies of the residues, it is unlikely that the entire set of dominant NRFs can be excited from only one target signature. As shown above, target signatures obtained from different aspects and polarization states excite a different subset of dominant NRFs. Certainly, the use of multiple target signatures obtained from multiple aspect and polarization states for target characterization allows us to retrieve at least a larger subset of dominant NRFs within the frequency bandwidth. The multiple-aspect multiple-polarization data set [49], a data set that consists of transient target signatures obtained with different transmit-receive configurations and polarization basis, is thus required. There are a number of possible ways to illuminate the target and post-process these target signatures, and we want to identify an efficient way to handle the data. To illustrate the different possible ways to handle such large data sets, an example of a simple human breast model shown in Figure 6(a) and (b), a lossless dielectric hemisphere with a small different dielectric spheres embedded, that mimics the breast cancer detection scenario [50, 51, 52, 53] is used. The radius of the lossless hemisphere is 60 mm with the relative permittivity of 5 (fat infiltrated tissue at ~3 GHz [54]). A 10 mm radius lossless dielectric sphere with a relative permittivity of 50 (taken from the Debye model [55] for <3 GHz) embedded inside the hemisphere is used to model the tumor. The target is illuminated using plane wave at six different aspects ($\theta ={105}^{°}$ and $\varphi ={30}^{°},{60}^{°},{90}^{°},{120}^{°},{150}^{°}$ and ${180}^{°}$, where $\theta$ is measured from the positive z-axis, while $\varphi$ is measured from the positive x-axis), resulted in 36 transmit-receive combinations (6 are monostatic, and 30 are bistatic) and 8 polarization states, i.e., 288 target signatures. It is apparent that the amount of data is tremendously increased when both aspect and polarization domains are considered. Efficient methods to handle the data are thus required.

Figure 6.

(a) Cross-sectional views (x-y plane) of the target. (b) Cross-sectional view ($\varphi ={30}^{°}$,$\theta ={105}^{°}$) of the breast volume under plane wave illuminations from ${\theta }_t={105}^{°}$, ${\varphi }_t={30}^{°}$, and ${\varphi }_t={210}^{°}$, respectively. (reprinted from [49] with permission from IEEE).

First, extraction results of the entire multiple-aspect multiple-polarization data sets with 288 target signatures (Case 1: $6\times 6$ aspects and eight polarizations) are tabulated in column 2 of Table 3. Six dominant resonant modes are extracted. This result is treated as a ground truth as the extraction has taken all the data into account. Next, we consider NRF extractions in linear and circular polarization bases separately with 144 target signatures (Case 2: 36 aspects and four polarization states) in each basis. All the six resonant modes are retrieved in both bases, and the corresponding boxes in columns 3 and 4 in Table 3 are shaded. The results indicate that both bases should give similar results in the NRF extraction process once all the four components in the Sinclair matrix are utilized.

Aspects	Case 1; All ($6\times 6$)	Case 2; All ($6\times 6$)		Case 3; All ($6\times 6$)
Pol.	Linear + circular; ($4+4$)	Linear; (4)	Circular; (4)	VV	HH	HV	VH	LL	RR	RL	LR
P	288	144	144	36	36	36	36	36	36	36	36
NRFs; $\frac{\sigma \pm \mathit{j\omega }}{c}$	$-3.48\pm j27.97$^1
	$-4.52\pm j32.93$^2
	$-2.13\pm j41.59$^3
	$-3.14\pm j45.27$^4
	$-3.27\pm j52.66$^5
	$-4.16\pm j57.63$^6

Table 3.

Comparison of the extracted natural resonance frequencies (NRFs) using multiple-aspect multiple-polarization data.

The shaded boxes indicate the NRF is properly retrieved in the extraction process (P, total number of target signatures; Reprinted from [49] with permission from IEEE)

Lastly, we perform NRF extraction on the multiple-aspect-only data at each of the eight polarization states (Case 3: 36 aspects and one polarization states). When HH data is used, five out of six resonant modes were extracted. When VV, HV, and VH data is used, only four modes were retrieved. When any one of the circularly polarized target signatures is used, all the six modes are accurately retrieved in all the co- and cross polarized results. The results show that we can retrieve all the dominant NRFs when only one of the four circularly polarized target signatures is used in the extraction process.

In summary, it is essential to include multiple-aspect data for target characterization because the NRF can be poorly excited at specific transmit-receive configurations [39, 48]. If all the four components in the scattering matrix are utilized for resonance extraction, both linear and circular polarization bases should give similar results. If only one out of the four components in the polarization matrix is utilized (regardless single or multiple aspects), circularly polarized components are preferred over linearly polarized components as some of the dominant modes may not be retrieved in certain linear polarization states. Compared to previous studies that require a large number of target signatures from multiple linear polarization angles (6 [45] and 18 [9]), the computational load is reduced to the maximum of four or even one without deteriorating the accuracies of the extracted NRF when polarimetric signature is utilized to handle the polarization dependency of the residues. The findings presented here provide us with guidelines on how we should illuminate the targets and process the multiple-aspect multiple-polarization data set to extract a set of NRF that includes all the dominant NRFs for target characterization. With the “completed” set of NRF, the E-pulse is constructed and then used for ATR. In most ATR scenarios in which only one target signature from the target is measured, the traditional E-pulse technique using the “completed” E-pulse should be able to effectively distinguish the correct targets from the others. In situations, where more than one target signatures from a target are considered for ATR, e.g., the situation of multidirectional E-pulse presented in [48], novel ATR procedures will be required to effectively utilize the “completed” set of NRF with the extensive data set.

3.3. Subsurface target detection

Other than target recognition problems with the target in free space, research activities have also focused on subsurface target detection—where the target is located at a particular depth below an interface. The motivation of subsurface target detection first originated from unexploded ordinance (UXO) detection using ground-penetrating radar [56, 57, 58, 59, 60] and later detection of tumors inside a breast volume [49, 51, 55, 61] as well as detection of small changes in hip prostheses [62, 63, 64].

In the subsurface target detection problem, usually the excitation and the measured field points are located in one medium, and the target is located in another medium. The scattered response is no longer solely dependent on the target itself but also interactions between the target and the interface. These interactions vary as the dielectric contrast between the two media, the depth and the orientation of the target [56, 57, 58, 59, 60], as well as the interactions between the heterogeneity of the medium and the target. In [65], a transient scattering of metallic targets sited below lossless and lossy half space is studied using joint TF analysis. All the interactions of the target and the dielectric interface, as well as attenuation phenomena due to the non-zero conductivity of the half space, are clearly observed in the joint TF domain. Considering a relatively simple situation where both media are homogeneous, studies have demonstrated that there are two types of resonances associated with the entire scattering problem—a target resonance and an image resonance. A target resonance is the NRF associated with the target attributes (geometry and the dielectric properties) and the dielectric properties of the medium in which the target is embedded [56, 57, 58, 59, 60]. An image resonance is the NRF that corresponds to the target depth—the distance between the target and the interface of the two media [56, 57]. For target detection and recognition applications, the target NRF is of interest. Studies [56, 57, 58, 63] showed that the target NRFs forms a spiral trajectory in the S-plane as the target depth changes. The spiral trajectory surrounds the target NRF within a homogeneous environment—the case where the target is fully immersed in the environment without a dielectric interface.

Other than the characterization of the target resonances for the subsurface target, attempts have also been made to apply the E-pulse for monitoring depth changes [64, 66] and geometrical changes [67] of a hip prosthesis model sited below a half space of tissue. Results show that the E-pulse technique is capable of detecting both depth changes and physical changes of the target. A subsurface target detection technique is proposed in which the E-pulse is constructed using the NRFs for a target inside a homogenous environment [60] to approximate the target NRFs for subsurface targets. This E-pulse is convolved with target signatures from the subsurface target and shows that this approximate technique can distinguish between different targets [34, 68, 69].

3.4. Other applications

A critical issue that affects the accuracies of the extracted NRFs is the commencement of the late time or the turn-on time of the resonant modes [30]. The damped exponential model given by Eq. (2) is strictly only valid during the late time period. The early time consists of high-frequency scattering centers that are mainly local scattering events, which can be modeled using the entire functions (e.g., a Gaussian [17]). The inclusion of the early time period into the NRF extraction process will undoubtedly degrade the accuracies of the retrieved NRFs. Automated detection of the commencement of late time without a priori knowledge of the target geometry or orientation becomes crucial for ATR. Hargrave et al. [70] proposed a method to estimate the commencement of the late time based on intrinsic differences between the full-rank Hankel matrix generated from the early time data and the rank-deficient late time matrix generated by discrete resonant components. Rezaiesarlak and Manteghi [71, 72] propose the short-time matrix pencil method (STMPM), which mostly applies the MPM for NRF extraction within a time window with a proper direction. The time window is moved by small time steps, and the extraction process is repeated until the entire signal is covered. The STMPM was first applied to chipless radio-frequency identification (RFID) application [71, 72, 73, 74]. Data is encoded as NRFs of the RFID tagged by incorporating notches in the structure such that each RFID tag has different NRFs [73, 74].

In addition to the applications above, the concept of late time resonances has also been applied to sensing applications in other disciplines. This includes monitoring the deployment of arterial stents [75], nondestructive evaluation of the maturity of fruit [76], automated detection of objects fallen on railway tracks [77], nondestructive evaluation of layered materials [78, 79], and detection of concealed handguns [80].

4. Conclusions and ongoing challenges

The fundamentals and development of resonance-based target recognition over the last 40 years have been briefly reviewed. Prospects of using polarimetric transient signatures for target classification and recognition have been demonstrated through numerical examples. Results show that the use of polarimetric signatures will undoubtedly enhance the ATR performance while reducing the amount of data to process.

Other than using a linear polarization basis to synthesize the circularly polarized target signature, the challenges for designing circularly polarized time-domain antennas [81] that directly generate circularly polarized pulses are an ongoing research topic for antenna engineers. In situations where the targets under test are very similar, the resonance-based target recognition would perform poorly and probably fails. Recent results in [51, 82, 83] have shown that incorporating polarimetric features, e.g., characteristic polarization states (CPS) of the resonance modes, can solve the problem. Development of novel target recognition schemes that utilize these novel feature sets would be of practical interest to the radar community.

In addition to defense and security applications, recent studies have been looking into the potential of applying the technology to nondestructive evaluations, medical diagnosis, RFID, and agricultural applications. Compared to radar applications where the target is isolated (e.g., aircrafts in the sky) and located in the far-field region and direct signal path from transmitting and receiving antennas exists higher-order interactions between the targets and antennas as well as scattering from surrounding objects become significant especially when the target is located in the near-field region in medical diagnosis [52, 53] and aforementioned RFID applications. Novel calibration procedures [84, 85] and signal processing solutions for NRF extractions of multiple targets [86, 87, 88] are still ongoing research topics for researchers to explore. Regarding subsurface target detection, our results demonstrate that the E-pulse is capable of detecting changes of the NRFs of subsurface targets due to depth [64, 66] and geometrical changes [67]. In conjunction with target NRF, other features embedded in the target signature, such as the “turn-on” time of the resonance [65], could also be used for these applications. Rather than solely relies on NRF for ATR, the trend of having a “combined” feature set with other parameters (e.g., CPS, turn-on time) will become the fashion for further development in this context.